Smoothed Particle Method Research Articles

Investigating landslide-induced tsunamis using a low-dissipation Riemann weakly compressible smoothed particle hydrodynamics model

In order to address the complex moving boundary problem and violent free-surface flows involved in the process of landslide tsunamis, this paper proposes a new low-dissipation Riemann-solver-based smoothed particle hydrodynamics (SPH) numerical model. In this model, the complex fluid-rigid slide interaction and friction effects on landslides are handled by an improved motion boundary approach based on a one-sided Riemann scheme. A weighted essentially non-oscillatory (WENO) reconstruction method is introduced to reduce excessive dissipation of the fluid caused by rigid landslide impact. The kernel gradient accuracy is assured by applying the corrected smoothed particle method (CSPM) into a continuity equation. Single-block and multi-block landslide tsunamis in two-dimensional (2D) or three-dimensional (3D) applications are simulated with different particle resolutions, and the results are compared to those of other numerical models and experimental results. The landslide trajectories obtained from the modified boundary model are in good agreements with the experimental values. Free-surface flows variations during the propagation are properly captured in the current simulations without introducing any artificial viscosity term, and the wave peaks exhibit deviations of −14%–10.5% from the experimental values, thus verifying the usability and robustness of the proposed Riemann-SPH model.

Fluid-structure interaction dynamic analysis of large civil aircraft tank sloshing

To meet the requirements of passengers, the tank volume of large civil aircraft is gradually increasing, which brings certain risks. Under different 0 pertaining conditions during the flight, the liquid in the tank will cause structural failure and affect the aircraft’s safety. Therefore, based on the assumption of an ideal fluid, the smooth particle method (SPH) combined with the finite element method (FEM) is used to establish the coupling dynamic equation between the tank and the liquid. The effects of sloshing impact on the water tank under different filling rates were studied, and the influences of the water wave shape, the center of gravity shift, stress distribution, and deformation of the tank were obtained. The results show that the mass accumulation of water particles occurs in the acceleration stage, the relative center of gravity changes greatly in less water, and the accumulated liquid greatly influences the tank under the action of inertia. The more liquid in the tank is, the more the impact effect on the tank increases. The analysis results provide some reference for the analysis and design of tank anti-sway plates and flight attitude adjustment.

Efficient and stable SAV-based methods for gradient flows arising from deep learning

The optimization algorithm plays an important role in deep learning and significantly affects the stability and efficiency of the training process, and consequently the accuracy of the neural network approximation. A suitable (initial) learning rate is crucial for the optimization algorithm in deep learning. However, a small learning rate is usually needed to guarantee the convergence of the optimization algorithm, resulting in a slow training process. We develop in this work efficient and energy stable optimization methods for function approximation problems as well as solving partial differential equations using deep learning. In particular, we consider the gradient flows arising from deep learning from the continuous point of view, and employ the particle method (PM) and the smoothed particle method (SPM) for the space discretization while we adopt SAV-based schemes, combined with the adaptive strategy used in Adam algorithm, for the time discretization. To illustrate the effectiveness of the proposed methods, we present a number of numerical tests to demonstrate that the SAV-based schemes significantly improve the efficiency and stability of the training as well as the accuracy of the neural network approximation. We also show that the SPM approach gives a slightly better accuracy than the PM approach. We further demonstrate the advantage of using adaptive learning rate in dealing with more complex problems.

A frictional contact algorithm in smoothed particle method with application in large deformation of soils

A frictional contact algorithm in smoothed particle method with application in large deformation of soils

Simulation of the dynamics of laser beams in an array of carbon nanotubes using the hydrodynamic approach

We simulated the propagation of a quasi-monochromatic laser beam in a medium with carbon nanotubes. Equations describing the dynamics of laser beams in an array of carbon nanotubes are obtained based on the hydrodynamic approach for the nonlinear Schrödinger equation. This equation is solved numerically using the smoothed particle method. The evolution of the beam is analyzed depending on the frequency of the electric field.

Influence of PPD and Mass Scaling Parameter on the Goodness of Fit of Dry Ice Compaction Curve Obtained in Numerical Simulations Utilizing Smoothed Particle Method (SPH) for Improving the Energy Efficiency of Dry Ice Compaction Process

The urgent need to reduce industrial electricity consumption due to diminishing fossil fuels and environmental concerns drives the pursuit of energy-efficient production processes. This study addresses this challenge by investigating the Smoothed Particle Method (SPH) for simulating dry ice compaction, an intricate process poorly addressed by conventional methods. The Finite Element Method (FEM) and SPH have been dealt with by researchers, yet a gap persists regarding SPH mesh parameters’ influence on the empirical curve fit. This research systematically explores Particle Packing Density (PPD) and Mass Scaling (MS) effects on the agreement between simulation and experimental outputs. The Sum of Squared Errors (SSE) method was used for this assessment. By comparing the obtained FEM and SPH results under diverse PPD and MS settings, this study sheds light on the SPH method’s potential in optimizing the dry ice compaction process’s efficiency. The SSE based analyses showed that the goodness of fit did not vary considerably for PDD values of 4 and up. In the case of MS, a better fit was obtained for its lower values. In turn, for the ultimate compression force FC, an empirical curve fit was obtained for PDD values of 4 and up. That said, the value of MS had no significant bearing on the ultimate compression force FC. The insights gleaned from this research can largely improve the existing sustainability practices and process design in various energy-conscious industries.

Application of Finite Element Technique: A Review Study

The finite element approach is used to solve a variety of difficulties, including well bore stability, fluid flow production and injection wells, mechanical issues and others. Geomechanics is a term that includes a number of important aspects in the petroleum industry, such as studying the changes that can be occur in oil reservoirs and geological structures, and providing a picture of oil well stability during drilling. The current review study concerned about the advancements in the application of the finite element method (FEM) in the geomechanical field over a course of century. Firstly, the study presented the early advancements of this method by development the structural framework of stress, make numerical computer solution for 2D thermal stress then stress analysis of the airplane. The second part focused on the most recent developments of FEM, and this method generates new techniques for solving these problems, such as the 1D, 2D, and 3D finite element models; the dynamic program method (DPM); the finite discrete element method (FDEM); and the finite element extended method (FEXM). The third part of this study presented the reservoir finite element simulation used for injection well testing inside unconsolidated oil sand reservoirs. Also improvement of the FE software program for the analyses, finite element extended approach to convert a 3D fault model were introduced. In addition, the study explored the development of a 3D and 4D model utilizing Visage for FEM analysis for geomechanics investigations, and the software eclipse for pressure drop prediction in carbonate reservoir weak formation and presented the Finite-Element Smoothed Particle Method (FESPM).

A meshless method coupling peridynamics with corrective smoothed particle method for predicting material failure

In this paper, we developed a meshless framework that couples peridynamics (PD) and the corrective smoothed particle method (CSPM) to simulate the complex damage and fracturing process of structures. This framework retains the advantages of PD in simulating material failure, and solves the problem that PD cannot simulate structures with irregular grids, and it also reduces the interface effect. Furthermore, the coupled PD-CSPM overcomes the challenge that stress boundary cannot be directly applied on CSPM-based models. In addition, PD-CSPM is capable of eliminating the limitation of Poisson's ratio imposed by the bond-based PD method. By simulating the deformation and failure behaviors of structures, the performance of PD-CSPM is verified and validated by the experimental observation and the numerical results obtained using other methods (the classic finite element method (FEM), extended finite element method (XFEM), and phase field method). Using the developed method, we successfully simulated fracture and failure behaviors of structures with irregular grids.

Application of the Godunov-Type Corrective Smoothed Particle Method to Impulsive Load Studies

The smoothed particle hydrodynamics (SPH) method is based on the kernel particle approximation, which is sensitive to the uniformity of the SPH particle distribution in the computational domain; that is, all SPH particles must be distributed evenly in the computational domain. These factors significantly influence the practical application of the SPH method. Meanwhile, calculating the sum near the boundaries of the computational domain may cause boundary defect problems since there are insufficient particles in the support domain, thus often resulting in relatively high errors in numerical simulation results near boundaries. To address these problems, the kernel particle approximation discrete process was corrected based on the traditional SPH method, and the corrected SPH method, the Godunov-type corrective smoothed particle method (CSPM), was formulated by introducing Riemann decomposition. In this study, the traditional SPH method and Godunov-type CSPM method were applied in a comparative study of discontinuous function problems, 1D shock tubes and 1D detonation waves. According to the analysis results, the Godunov-type CSPM method can not only effectively improve the calculation accuracy and compatibility of the traditional SPH method in discontinuous shock wave problems but also increase the accuracy of the traditional SPH method in capturing strong discontinuities.

Nonrelativistic and relativistic simple wave problems solved with Smoothed Particle Hydrodynamics

To simulate the expansion of the matter created in relativistic nuclear collisions, codes in 3+1 dimensions are used and we are developing a new one. To benchmark such codes, the Sod’s shock tube is often used. A closely related problem is the one-dimensional expansion of a gas into vacuum. In this paper, we study this problem classically and relativistically with the Smoothed Particle Method and test various techniques to improve the precision and speed of the solution.

Simulation Prediction and Forming Quality Control of Hemming Structure with Dissimilar Materials Oriented to Process Chain

Multi-material mixed-use has become an important way to realize autobody lightweight. However, due to the difference of material properties, the structure is prone to material mismatch when using thin and dissimilar materials. In this paper, a deformation compensation method oriented to the process chain is proposed to eliminate the quality defects. Based on the simulation and prediction of the process chain, the final deformation is advanced to the spreading preparation stage to realize the geometric pre-compensation of the structure. Firstly, in the spreading and hemming stage, the prediction of hemming with adhesive was realized by establishing SPH (Smoothed particle method)-FEM (Finite element method) coupling model. Then, a coupled thermal-chemical-structural model is established to predict the curing deformation. Finally, the deformed dimension is fed back to the spreading stage to achieve the deformation compensation. The results show that the multi-field coupling model can achieve accurate prediction, and the structural deformation is reduced to more than 95%. This provides a basis for material matching and optimization of the whole process chain.

Vibrating behavior of submerged floating tunnel in current field investigated with hybrid vector-autoregressive model

This paper presents a study for identifying damage severity and location in the supporting system of tether-type submerged floating tunnel (SFT) as an innovative manner for crossing a variety of water area. A novel sensitive damage detection approach is proposed in the context of statistical pattern recognition to analyze the vibrating behaviors of the main tube and the tether system of SFT induced by currents acting upon both the tube and the tethers by making use of the time-domain displacement and force series obtained from numerical simulations based on smoothed particle method (SPH). From clustering of the constructed potential damage indicators fitted to SFT responses in currents, different damage conditions can be recognized qualitatively with the severity quantified by two clustering indices. Clustering patterns of the damage indicators provides the possibility of identifying damages in a certain tether when missing the direct force records of the tether in question. The damage information can be detected by the locations and separations of the probability density distribution curves based on the Mahalanobis distances generated from the damage indicators. An insight into the antagonistic and balancing effects between the tethers in the supporting system of SFT can also be taken with the Mahalanobis distance distributions.

A Mindlin shell model based on the corrective smoothed particle method and accuracy implementation of the free boundary

A meshless shell method for large deformation and a linear relation between the Cauchy stress tensor and the Almansi strain tensor based on the corrective smoothed particle method (CSPM) is presented in this paper. Due to the use of the shell theory, only one layer of particles in the reference plane is required to discretize the shell model. The CSPM combining the kernel estimate with the Taylor series expansion is adopted, which resolves the general problems of low precision and particle deficiency in standard smoothed particle hydrodynamics (SPH). The discrete governing equations of the shell in the strong form are derived using the conservation condition and the CSPM interpolation function. Aiming at the sore point of the free boundary in the meshless method, the developed model enables the modified governing equations to automatically satisfy the free boundary condition without additional treatment. Moreover, the total Lagrangian kernel function and stress points are employed to eliminate tensile instability and instability induced by the rank deficiency. Finally, several numerical examples are used to verify the validity and accuracy of the meshless shell model.

A three-dimensional beam formulation for large deformation and an accurate implementation of the free boundary

This paper presents a meshless model for quasi-static and dynamic analysis of a three-dimensional Timoshenko beam with geometric nonlinearity. A general mathematical formulation is constructed based on the corrective smoothed particle method (CSPM), which can correct the low precision and completeness deficiency of the standard smoothed particle hydrodynamics(SPH) method. The discrete governing equations as well as the boundary conditions in strong form for the three-dimensional beam are then derived by using the conservation conditions and the CSPM interpolation function. The developed model enables one to expediently discretize the geometric nonlinear beam with only a row of particles at the central axis and to automatically satisfy the free boundary condition without any additional treatment. Moreover, Lagrangian kernel function and stress points are adopted to eliminate tensile instability and instability induced by the rank deficiency within the particle methods. Finally, comparisons with several results obtained from the existing literature are provided to demonstrate the validity and potential of the present procedure.

Mathematical modeling the process of wire surfacing by the smoothed particle hydrodynamics method

We discuss issues the process of layer-by-layer synthesis of metal products by wire surfacing. A mathematical model of the process has been developed and implemented. The model allows one to calculate the volumetric distributions of temperatures, melt flow rates, pressures, components of heat fluxes density, the shape and dimensions of the molten bath, the shape of the free surface of the molten metal, the shape and dimensions of the weld bead. For this purpose, the main physical processes that affect the formation of the metal product were considered, namely: melting and crystallization of the metal, surface tension, the Marangoni effect and the heat source features. The smoothed particle method was used to implement this model. A number of numerical experiments were carried out using the developed model. Firstly, the model parameters for steel and titanium were identified and verified. After that, calculations were carried out to verify the model itself using the obtained roller geometry. The numerical solution for the deposition of one layer was compared with the full-scale experiment. The error was no more than 5% in any direction. Thus, the efficiency and correctness of the constructed model is shown.

Machine learning from a continuous viewpoint, I

We present a continuous formulation of machine learning, as a problem in the calculus of variations and differential-integral equations, in the spirit of classical numerical analysis. We demonstrate that conventional machine learning models and algorithms, such as the random feature model, the two-layer neural network model and the residual neural network model, can all be recovered (in a scaled form) as particular discretizations of different continuous formulations. We also present examples of new models, such as the flow-based random feature model, and new algorithms, such as the smoothed particle method and spectral method, that arise naturally from this continuous formulation. We discuss how the issues of generalization error and implicit regularization can be studied under this framework.

Применение стационарной гипопластической модели для моделирования движения сыпучей среды

Для моделирования движения гранулированной среды предложен вариант стационарной гипопластической модели. С целью решения задачи выбран метод сглаженных частиц, особенностью реализации которого является использование в расчетах комбинированного ядра, с малым параметром сглаживания. Выполнен расчет процесса обрушения песчаного столба и проведено сравнение полученных численных результатов моделирования с известными экспериментальными данными. Purpose. The aim of this study is а development of mathematical models that describe the complex motion of a granular medium in the process of its disintegration that allows evaluating the possibility of using a simplified stationary hypoplastic model to describe this process. Methodology. To describe the motion of a granular medium, the classical equations of motion and mass conservation were used. The calculation of the deviator of the stress tensor is performed within the framework of a simplified hypoplastic model. The work uses the hypothesis of a linear relationship between the pressure function and the density of the granular medium. This hypothesis is typical for smoothed particle methods, one of the variants of which is proposed for the numerical implementation of the problem. Results. A new mathematical model of the problem of the motion of a granular medium in the process of its disintegration is formulated. An algorithm for solving the problem based on the method of smoothed particles is proposed. The problem of disintegration of a sand pillar is solved numerically. A comparative analysis of the obtained solutions with experimental data is accomplished. Findings. Using the assumption that the propagation velocity of elastic waves that determine the stress-strain state in sand particles is much higher than the velocities of particles arising from sand shedding, an approximate model for calculating the stress state of a moving granular medium based on a stationary hypoplastic model is presented. To solve the problem, the method of smoothed particles with a small smoothing parameter was implemented with the help of a new combined interpolation core in the calculations. To verify the mathematical model and the selected method, the implementation of the smoothed particle method was performed and the problem of disintegration of the sand column was solved. The obtained solution of the problem shows satisfactory agreement with the experimental data.

Study on coupling of smooth particle method with finite elements based on mesh–particle matching degree

Study on coupling of smooth particle method with finite elements based on mesh–particle matching degree

Smoothed Particle Method for Full-Wave Simulation of Microwave Applicators With Translational-Rotary Elements

Elements in both translation and rotation such as the screw propeller are widely applied to improve microwave heating uniformity. However, it is challenging to numerically model this heating process due to the difficulties in dynamic meshing for the moving elements. Our previous transformation optics method will be also burdensome to use in this case because it is very difficult to get the permittivity and permeability tensors. In this letter, a mesh-free numerical calculation based on a smoothed particle method is proposed to solve this complicated case. The proposed method uses smoothed particles to replace the meshes. Thus, the motion of the elements can be replaced by the motion of the particles and the dynamic meshing process can be avoided. A 2-D model is computed and the results are compared with those calculated by the finite element method. Good agreements are achieved.

Pebble-driven planet formation for TRAPPIST-1 and other compact systems

Recently, seven Earth-sized planets were discovered around the M-dwarf star TRAPPIST-1. Thanks to transit-timing variations, the masses and therefore the bulk densities of the planets have been constrained, suggesting that all TRAPPIST-1 planets are consistent with water mass fractions on the order of 10%. These water fractions, as well as the similar planet masses within the system, constitute strong constraints on the origins of the TRAPPIST-1 system. In a previous work, we outlined a pebble-driven formation scenario. In this paper we investigate this formation scenario in more detail. We used a Lagrangian smooth-particle method to model the growth and drift of pebbles and the conversion of pebbles to planetesimals through the streaming instability. We used the N-body code MERCURY to follow the composition of planetesimals as they grow into protoplanets by merging and accreting pebbles. This code is adapted to account for pebble accretion, type-I migration, and gas drag. In this way, we modelled the entire planet formation process (pertaining to planet masses and compositions, not dynamical configuration). We find that planetesimals form in a single, early phase of streaming instability. The initially narrow annulus of planetesimals outside the snowline quickly broadens due to scattering. Our simulation results confirm that this formation pathway indeed leads to similarly-sized planets and is highly efficient in turning pebbles into planets. Our results suggest that the innermost planets in the TRAPPIST-1 system grew mostly by planetesimal accretion at an early time, whereas the outermost planets were initially scattered outwards and grew mostly by pebble accretion. The water content of planets resulting from our simulations is on the order of 10%, and our results predict a “V-shaped” trend in the planet water fraction with orbital distance: from relatively high (innermost planets) to relatively low (intermediate planets) to relatively high (outermost planets).